Volume 28, Number 3
Volume 28, Number 3, 2023
Kozlov V. V.
Abstract
An example of an analytic system of differential equations in $\mathbb{R}^6$ with an equilibrium
formally stable and stable for most initial conditions is presented. By means of a divergent formal transformation this system is reduced to a Hamiltonian system with three degrees of freedom. Almost all its phase space is foliated by threedimensional invariant tori carrying quasiperiodic trajectories.
These tori do not fill all phase space. Though the ``gap'' between these tori has zero measure, this set is everywhere dense in $\mathbb{R}^6$ and unbounded phase trajectories are dense in this gap. In particular, the formally stable equilibrium is Lyapunov unstable. This behavior of phase trajectories is quite consistent with the diffusion in nearly integrable systems. The proofs are based on the Poincaré–Dulac theorem, the theory of almost periodic functions, and on some facts from the theory of inhomogeneous Diophantine approximations. Some open problems related to the example are presented.

Cincotta P., Giordano C., Simó C.
Abstract
In this work an exhaustive numerical and analytical investigation of the dynamics
of a biparametric symplectic map, the socalled rational standard map, at moderatetolarge
values of the amplitude parameter is addressed. After reviewing the model, a discussion
concerning an analytical determination of the maximum Lyapunov exponent is provided
together with thorough numerical experiments. The theoretical results are obtained in the
limit of a nearly uniform distribution of the phase values. Correlations among phases lead to
departures from the expected estimates. In this direction, a detailed study of the role of stable
periodic islands of periods 1, 2 and 4 is included. Finally, an experimental relationship between
the Lyapunov and instability times is shown, while an analytical one applies when correlations
are irrelevant, which is the case, in general, for large values of the amplitude parameter.

Grines V. Z., Mints D. I.
Abstract
In P.D.McSwiggen’s article, it was proposed Derived from Anosov type construction
which leads to a partially hyperbolic diffeomorphism of the 3torus. The nonwandering set
of this diffeomorphism contains a twodimensional attractor which consists of onedimensional
unstable manifolds of its points. The constructed diffeomorphism admits an invariant onedimensional
orientable foliation such that it contains unstable manifolds of points of the
attractor as its leaves. Moreover, this foliation has a global cross section (2torus) and defines
on it a Poincar´e map which is a regular Denjoy type homeomorphism. Such homeomorphisms
are the most natural generalization of Denjoy homeomorphisms of the circle and play an
important role in the description of the dynamics of aforementioned partially hyperbolic
diffeomorphisms. In particular, the topological conjugacy of corresponding Poincaré maps
provides necessary conditions for the topological conjugacy of the restrictions of such partially
hyperbolic diffeomorphisms to their twodimensional attractors. The nonwandering set of
each regular Denjoy type homeomorphism is a Sierpiński set and each such homeomorphism
is, by definition, semiconjugate to the minimal translation of the 2torus. We introduce a
complete invariant of topological conjugacy for regular Denjoy type homeomorphisms that is
characterized by the minimal translation, which is semiconjugation of the given regular Denjoy
type homeomorphism, with a distinguished, no more than countable set of orbits.

Popelensky T. Y.
Abstract
For two dimensional surfaces (smooth) Ricci and Yamabe flows are equivalent. In
2003, Chow and Luo developed the theory of combinatorial Ricci flow for circle packing metrics
on closed triangulated surfaces. In 2004, Luo developed a theory of discrete Yamabe flow for
closed triangulated surfaces. He investigated the formation of singularities and convergence to
a metric of constant curvature.
In this note we develop the theory of a naïve discrete Ricci flow and its modification — the socalled weighted Ricci flow. We prove that this flow has a rich family of first integrals and is equivalent to a certain modification of Luo’s discrete Yamabe flow. We investigate the types of singularities of solutions for these flows and discuss convergence to a metric of weighted constant curvature. 
Carvalho A. C., Araujo G. C.
Abstract
In this study, we analyze a planar mathematical pendulum with a suspension point
that oscillates harmonically in the vertical direction. The bob of the pendulum is electrically
charged and is located between two wires with a uniform distribution of electric charges, both
equidistant from the suspension point. The dynamics of this phenomenon is investigated. The
system has three parameters, and we analyze the parametric stability of the equilibrium points,
determining surfaces that separate the regions of stability and instability in the parameter
space. In the case where the parameter associated with the charges is equal to zero, we obtain
boundary curves that separate the regions of stability and instability for the Mathieu equation.
